1. MATHPOWER TM 12, WESTERN EDITION 4.1.1 Chapter 4 Trigonometric Functions 4.1

2. 4.1.2 An angle is formed by rotating an initial arm about a fixed point. Angles in Standard Position - Definitions An angle is said to be in standard position when the initial arm is on the positive x-axis and the vertex is at (0, 0). Positive angles have a counterclockwise rotation. Negative angles have a clockwise rotation. A principal angle is the angle measured from the positive x-axis to the terminal arm. The principal angle is always a positive measure. A reference angle is the angle measured from the closest x-axis to the terminal arm. The reference angle is always an acute angle and is always positive. Coterminal angles are angles that share the same terminal arm.

3. 4.1.3 Sketching Angles in Standard Position Sketch the following angles in standard position. State the principal angle, the reference angle, and one positive and one negative coterminal angle. a) 150 0 b) -260 0 c) 560 0 Principal Angle Reference Angle Coterminal Angles Principal Angle Reference Angle Coterminal Angles Principal Angle Reference Angle Coterminal Angles 150 0 30 0 510 0 -210 0 100 0 80 0 460 0 -620 0 200 0 20 0 560 0 -160 0

4. d) -220 0 Principal Angle Reference Angle Coterminal Angles 140 0 40 0 500 0 -580 0  = 140 + 360n where n is an element of the integers. Sketching Angles in Standard Position To find all coterminal angles: 4.1.4

5. Radian Measure A radian is the measure of the angle at the centre of the circle subtended by an arc equal in length to the radius of the circle. r r r 2r2r r r 4.1.5 number of radians =

6. r Changing Degree Measure to Radian Measure 4.1.6 Therefore, 2  rad = 360 0. Or,  rad = 180 0.

7. 4.1.7 Changing Degree Measure to Radian Measure Calculate the radian measure: a) 210 0 180 0  =  rad = 3.67 Exact radians Approximate radians b) 315 0 radians

8. 4.1.8 Changing Radian Measure to Degree Measure Calculate the degree measure: a)  rad = 180 0 = 120 0 b) = 15 0 c) 1.68 rad = 96.26 0 To convert from radians to degrees, multiply by To convert from degrees to radians, multiply by

9. 4.1.9 Finding the Sector Angle or the Arc Length Find the measure of the sector angle: 6.1 cm 5 cm Find the arc length: 8 cm 70 0 Convert 70 0 to radians: 18a = 56   a = 9.77 The arc length is 9.77 cm. a

10. Angular Velocity Radians are often involved in applications involving angular speed. Angular speed is the rate at which the central angle is changing. Find the average angular speed of a wheel that is rotating 15 times in 3 s. Each time that the wheel rotates, it turns through a central angle of 2  radians. Therefore, in 15 rotations, the wheel has travelled 30  radians. speed = 10  rad/s The average angular speed is 10  rad/s. 4.1.10 speed =

11. Suggested Questions: Pages 189-191 1, 2, 3-71 odd, 74, 80, 81, 83 4.1.11